On a new class of abstract impulsive functional differential equations of fractional order

Research Article

On a new class of abstract impulsive functional differential equations of fractional order

Pradeep Kumar^a,∗, Dwijendra N. Pandey^b, D. Bahuguna^a

aDepartment of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India.

bDepartment of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India.

Communicated by C. Cuevas

Abstract

In this paper, we prove the existence and uniqueness of mild solutions for the impulsive fractional differential equations for which the impulses are not instantaneous in a Banach spaceH. The results are obtained by using the analytic semigroup theory and the fixed points theorems. c2014 All rights reserved.

Keywords: Impulsive fractional differential equations, Analytic semigroup, Fixed point theorems.

2010 MSC: 34K45, 34A60, 35R12, 45J05.

1. Introduction

Fractional calculus refers to integration or differentiation of any (i.e., non-integer) order. The field has a history as old as calculus itself, which did not attract enough attention for a long time. In the past decades, the theory of fractional differential equations has become an important area of investigation because of its wide applicability in many branches of physics, economics and technical sciences. For a nice introduction, we refer to the reader to [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and references cited therein.

Impulsive effects are common phenomena due to short-term perturbations whose duration is negligible in comparison with the total duration of the original process. Such perturbations can be reasonably well approximated as being instantaneous changes of state, or in the form of impulses. The governing equations of such phenomena may be modeled as impulsive differential equations. In recent years, there has been a growing interest in the study of impulsive differential equations as these equations provide a natural frame

∗Corresponding author

Email addresses: [email protected](Pradeep Kumar),[email protected](Dwijendra N. Pandey), [email protected](D. Bahuguna)

Received 2013-04-30

work for mathematical modelling of many real world phenomena, namely in the control theory, physics, chemistry, population dynamics, biotechnology, economics and medical fields.

Due to the great development in the theory of fractional calculus and impulsive differential equations as well as having wide applications in several fields. Recently, the study of fractional differential equations with impulses has been studied by many authors (see [17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38]).

In [16], the authors introduced a new class of abstract differential equations for which the impulses are not instantaneous and investigated the existence of mild and classical solutions for the following system:

u⁰(t) =Au(t) +f(t, u(t)), t∈(s_i, t_i+1], i= 0,1,· · · , N, (1.1) u(t) =gi(t, u(t)), t∈(ti, si], i= 1,2,· · ·, N, (1.2)

u(0) =u0, (1.3)

whereA is the infinitesimal generator of aC0-semigroup of bounded linear operators,{S(t), t≥0}on a Banach space (H,k.k), the functions g_i∈C((t_i, s_i]×H;H) for eachi= 1,2,· · · , N andf : [0, T₀]×H→H is suitable function.

Motivated by the work [16], In this article, we consider the following impulsive fractional differential equations in a Banach space (H,k.k) for which impulses are not instantaneous:

CD^β_tu(t) +Au(t) = f(t, u(t), u(g(t))), t∈(s_i, t_i+1], i= 0,1,· · · , N, (1.4) u(t) = hi(t, u(t)), t∈(ti, si], i= 1,2,· · ·, N, (1.5)

u(0) = u0 ∈H, (1.6)

where^CD^β_t is the Caputo fractional derivative of orderβ,−Ais the infinitesimal generator of an analytic semigroup of bounded linear operators,{S(t), t≥0} on a Banach spaceH, the impulses start suddenly at the points t_i and their action continues on the interval [t_i, s_i], 0 = t₀ = s₀ < t₁ ≤ s₁ ≤ t₂ <, ..., < t_N ≤ sN ≤< t_N+1 = T0, the functions hi ∈ C((ti, si]×H;H) for each i = 1,2,· · ·, N, g : [0, T0] → [0, T0] and f : [0, T₀]×H×H →H are suitable functions.

The paper is organized as follows. In “Preliminaries and Assumptions” section, we provide some basic definitions, notations, lemmas and proposition which are used throughout the paper. In “Existence of mild solutions” section, we will prove some existence and uniqueness results concerning the PC-mild solutions.

In the last (i.e., In “Application”) section, we give an example to demonstrate the application of the main results.

2. Preliminaries and assumptions

In this section, we will introduce some basic definitions, notations, lemmas and proposition which are used throughout this paper.

It is assume that −A generates an analytic semigroup of bounded operators, denoted by S(t), t≥0. It is known that there exist constants ˜M ≥1 andω ≥0 such that

kS(t)k ≤M e˜ ^ωt, t≥0.

If necessary, we may assume without loss of generality that kS(t)k is uniformly bounded by M, i.e., kS(t)k ≤M fort≥0,and that 0∈ρ(−A),implies −A is invertible. In this case, it is possible to define the fractional powerA^α for 0≤α≤1 as closed linear operator with domain D(A^α)⊆H. Furthermore,D(A^α) is dense inH and the expression

kxk_α =kA^αxk,

defines a norm on D(A^α). Henceforth, we denote the space D(A^α) by Hα endowed with the norm k · k_α. Also, for eachα >0,we defineH−α= (Hα)^∗,the dual space ofHα, is akxk_−α =kA^−αxk.For more details, we refer to the reader to the book by Pazy [1, pp. 69].

Lemma 2.1. [1, pp. 72,74,195-196]Suppose that−Ais the infinitesimal generator of an analytic semigroup S(t), t≥0 with kS(t)k ≤M for t≥0 and 0∈ρ(−A). Then we have the following:

(i) H_α is a Banach space for 0≤α≤1;

(ii) For any 0< δ≤α impliesD(A^α)⊂D(A^δ), the embedding H_α ,→H_δ is continuous;

(iii) The operatorA^αS(t) is bounded, i.e., there exists a constantN such that kA^αS(t)k ≤N

and

kA^αS(t)k ≤Cαt^−α for each t >0.

Lemma 2.2. [16, Lemma 1.1] A set B ⊆ PC(Hα) is relatively compact in PC(Hα) if and only if set Bei is relatively compact in C([t_i, t_i+1];H_α]), where PC(H_α) is the space of piecewise continuous functions from [0, T₀] into H_α to be specified later.

Definition 2.3. [37, Def. 2.7] By the mild solution of the following system

CD^β_tu(t) +Au(t) = h(t), t∈[t0, T0], (2.1)

u(t₀) = u₀, (2.2)

we mean a continuous function u: [t₀, T₀]→H which satisfies the following integral equation u(t) =T(t−t0)u0+

Z t t0

(t−s)^β−1P(t−s)h(s)ds, t∈[t0, T0], where

T(t) = Z ∞

0

ξ_β(θ)S(t^βθ)dθ, P(t) =β Z ∞

0

θξ_β(θ)S(t^βθ)dθ, ξ_β(θ) = 1

βθ^−1−β1ρ_β(θ^−1β)≥0, ρ_β(θ) = 1

π

∞

X

n=1

(−1)ⁿ⁻¹θ^−nβ−1Γ(nβ+ 1)

n! sin(nπβ), θ∈(0,∞), ξ_β is a probability density function defined on (0,∞),that is

ξ_β(θ)≥0, θ∈(0,∞), Z ∞

0

ξ_β(θ) = 1,

and Z ∞

0

θ^γξ_β(θ) = Z ∞

0

1

θ^γβρ_β(θ) = Γ(1 +γ)

Γ(1 +γβ), for any γ ∈[0,1].

Lemma 2.4. The operators T(.) and P(.) have the following properties:

(i). {T(t), t≥0} and{P(t), t≥0} are strongly continuous.

(ii). If {S(t), t >0} is compact, then T(t) and P(t) are also compact operators for every t >0.

(iii). For any fixed t≥0, T(t) andP(t) are linear and bounded operators, i.e., for any x∈H, kT(t)xk ≤Mkxk and kP(t)xk ≤ βM

Γ(1 +β)kxk.

Remark 2.5. SinceT(.) andP(.) are associated with theβ, there are no analogue of the semigroup property, i.e., T(t+s)6=T(t)T(s), P(t+s)6=P(t)P(s) for t, s >0.

3. Existence of mild solutions

In this section, we prove the existence of mild solutions for the impulsive system (1.4)-(1.6). To begin, we use the following definition.

Definition 3.1. [16, Def. 2.1] A functionu∈ PC(H) is said to be a mild solution of the problem (1.4)-(1.6) ifu(0) =u₀, u(t) =h_i(t, u(t)) for allt∈(t_i, s_i], for eachi= 1,· · · , N and

u(t) =

Z ∞ 0

ξ_β(θ)S(t^βθ)u0dθ

+ β

Z t 0

Z ∞ 0

θξβ(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθds, for all t∈[0, t₁] and

u(t) =

Z ∞ 0

ξβ(θ)S((t−si)^βθ)hi(si, u(si))dθ

+ β

Z t si

Z ∞ 0

θξ_β(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθds, for all t∈[si, ti+1], for each i= 1,· · ·, N.

We define the set of functions as follows

PC(H_α) ={u: [0, T₀]→H_α :u(.) is continuous at t6=t_i, u(t⁻_k) =u(t_k), u(t⁺_k) exists for all i= 1,2,· · · , N}.

PC(H_α) is a Banach space endowed with the supremum norm kuk_PC:= sup

t∈I

ku(t)k_α. Now, we define the functions uei ∈C([ti, ti+1];Hα) given by

eui(t) =

u(t), fort∈(t_i, t_i+1], u(t⁺_i ), fort=ti

Let B⊆ PC(H_α), we define

Bei ={uei:u∈B}.

We shall use the following conditions onf and hi in its arguments:

(H1) LetW ⊂Dom(f) be an open subset ofR+×H_α×H_α, where 0≤α <1. For each (t, u, v)∈W, there is a neighborhood V1 ⊂W of (t, u, v), such that the nonlinear mapf :R+×Hα×Hα → H satisfies the following condition,

kf(t, u, v)−f(t, u1, v1)k ≤Lf{ku−u1k_α+kv−v1k_α} for all (t, u, v), (t, u₁, v₁)∈V₁, L_f =L_f(t, u, v, V₁)>0 is a constant.

(H2) Letg: [0, T₀]→[0, T₀] is continuous and satisfies the delay property g(t)≤t fort∈[0, T₀].

(H3) The functionsh_i : [t_i, s_i]×H_α→H_α are continuous and there are positive constantsL_hi such that kh_i(t, x)−h_i(t, y)k_α ≤L_hikx−yk_α,

for allx, y∈Hα, t∈[ti, si] and each i= 0,1,· · · , N.

(H4) Foru, v∈H_α, the functionf(., u, v) is strongly measurable on [0, T₀] andf(t, ., .)∈C(H_α×H_α, H)∈ fort∈ [0, T0]. There exists a constant β1 ∈[0, β) and mf ∈L

1

β1([0, T0],R⁺) such that kf(t, u, v)k ≤ m_f(t) for all (t, u, v)∈[0, T0]×Hα×Hα.

Theorem 3.2. Suppose the assumptions (H1)-(H3) hold and L= maxn

M L_hi+ 2CαL_fΓ(2−α)

(1−α)Γ(1 +β(1−α))T₀^β(1−α): i= 1,· · ·, No

<1.

(3.1) Then there exists a unique mild solution u∈ PC(H_α) of the problem (1.4)-(1.6).

Proof. Let us define a map Υ : PC(H_α) → PC(H_α), given by Υu(0) = u₀, Υu(t) = h_i(t, u(t)) for t ∈ (ti, si], i= 1,2,· · ·, N and

Υu(t) =

Z ∞ 0

ξβ(θ)S(t^βθ)u0dθ

+ β

Z t 0

Z ∞ 0

θξ_β(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθds, for all t∈[0, t₁] and

Υu(t) =

Z ∞ 0

ξ_β(θ)S((t−s_i)^βθ)h_i(s_i, u(s_i))dθ

+ β

Z t si

Z ∞ 0

θξ_β(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθds,

t∈[s_i, t_i+1], i= 1,2,· · · , N. (3.2) Clearly, Υ is well defined.

Next we show that Υ is contraction on PC(H_α).

Let φ, ψ∈ PC(H_α), i∈ {1,· · · , N} and t∈[s_i, t_i+1], we have

kΥφ(t)−Υψ(t)k ≤ k Z ∞

0

ξ_β(θ)kS((t−s_i)^βθ)k kh_i(s_i, φ(s_i))−h_i(s_i, ψ(s_i))k_αds

+ β

Z t si

Z ∞ 0

θξ_β(θ)(t−s)^β−1kA^αS((t−s)^βθ)k

×kf(s, φ(s), φ(g(s)))−f(s, ψ(s), ψ(g(s)))kdθds

≤ M L_hikφ−ψk_PC+CαLfβΓ(2−α) Γ(1 +β(1−α))

Z _t

si

(t−s)^{β(1−α)−1}[kφ(s)−ψ(s)k_α +kφ(g(s))−ψ(g(s))k_α]ds

≤ [M Lhi+ 2C_αL_fΓ(2−α)

(1−α)Γ(1 +β(1−α))T₀^β(1−α)]kφ−ψk_PC.

Thus, we have

kΥφ−Υψk_{C([si,ti+1];Hα)}≤[M Lhi+ 2C_αL_fΓ(2−α)

(1−α)Γ(1 +β(1−α))T₀^β(1−α)]kφ−ψkPC.

(3.3) Continuing in this fashion, we get

kΥφ−Υψk_{C([0,t1];Hα)}≤ 2C_αL_fΓ(2−α)

(1−α)Γ(1 +β(1−α))T₀^β(1−α)kφ−ψk_PC, (3.4) kΥφ−Υψk_C([s

i,ti];Hα)≤L_hikφ−ψk_PC, i= 1,· · ·, N. (3.5) Hence, from (3.3)-(3.5), we have

kΥφ−Υψk_PC ≤Lkφ−ψk_PC,

i.e., Υ(.) is a contraction and there exists a unique mild solution of (1.4)-(1.6).

The next result concerning the existence and uniqueness of mild solutions for the impulsive system (1.4)-(1.6) under the assumptions (H3) and (H4).

Theorem 3.3. Suppose the assumptions (H3)and (H4) hold. The semigroup{S(t);t≥0} is compact, the functions h_i(.,0)are bounded, (3.1) holds and for each u₀ ∈H_α, letr >1 and 0< δ <1 be such that











Mku₀k_α+ (1 +M) maxi=1,···,Nkh_i(.,0)k_α ≤ (1−δ)r, maxi=1,···,N

n

Lhi(1 +M)kuk_PC+Cα Γ(2−α) Γ(1+β(1−α))

T₀^β(1−α) 1−α ℵo

≤ δr,

CαΓ(2−α) Γ(1+β(1−α))

T₀^β(1−α)

1−α sup_{s∈[0,t1],v∈Br(0,PC(Hα))}kf(s, v(s), v(g(s)))k ≤ δr,

(3.6)

where ℵ= sup_{s∈[si,ti+1],v∈Br(0,PC(Hα))}kf(s, v(s), v(g(s)))k.

Then there exists a mild solution u∈ PC(H_α) of the impulsive problem (1.4)-(1.6).

Proof. Let Υ =

N

X

i=0

Υ¹_i +

N

X

i=0

Υ²_i, where Υ^j_i :PC(H_α)→ PC(H_α), i= 0,1,· · ·, N, j = 1,2, are given by

Υ¹_iu(t) =























h_i(t, u(t)), for t∈(t_i, s_i], i≥1, R∞

0 ξ_β(θ)S((t−s_i)^βθ)h_i(s_i, u(s_i))ds, for t∈(s_i, t_i+1], i≥1, 0, for t /∈[ti, ti+1], i≥0, R∞

0 ξ_β(θ)S(t^βθ)u₀dθ, for t∈[0, t₁], i= 1,

Υ²_iu(t) =









 βRt

si

R∞

0 θξ_β(θ)(t−s)^β−1S((t−s)^βθ)

×f(s, u(s), u(g(s)))dθds, for t∈(s_i, t_i+1], i≥0, 0, for t /∈(si, ti+1], i≥0.

Also, letλ= _1−β^β−1

1 ∈(−1,0), b= ^{β(1−α)−1}_1−β

1 >0, M1 =km_fk

L

1

β1[0,T0],where β1 ∈[0, β), 0< α≤1.

It is easy to see that (t−s)^β−1 ∈L^1−β11[0, t], for t∈[0, T₀].

By using H¨older inequality and (H4), for t∈[t1, t2]⊆[0, T0], we get Z t

t1

|(t−s)^β−1f(s, u(s), u(g(s)))|ds ≤ Z t t1

(t−s)^λds1−β₁

km_fk

L

1 β1[t1,t]

≤ M1

(1 +λ)^1−β1T₀^{(1+λ)(1−β1)}. Then, we have

β Z t

t1

Z ∞ 0

θξβ(θ)|(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))|dθds

≤M β Z t

t1

Z ∞ 0

θξ_β(θ)|(t−s)^β−1f(s, u(s), u(g(s)))|dθds

≤ M β Γ(1 +β)

Z t t1

|(t−s)^β−1f(s, u(s), u(g(s)))|ds

≤ M β Γ(1 +β)

Z t t1

(t−s)^λds 1−β1

km_fk

L

1 β1[t1,t]

≤ βM M1

Γ(1 +β)(1 +λ)^1−β1T₀^{(1+λ)(1−β1)}. (3.7)

Our aim is to prove that the map Υ is a condensing map from Br(0,PC(Hα)) into Br(0,PC(H_α)). For this we have divided our proof into four steps.

Step 1. First we show that ΥB_r(0,PC(H_α))⊂B_r(0,PC(H_α)),where B_r(0,PC(H_α)) ={u∈ PC(H_α) :kuk_α ≤r}, forr >0.

Let u∈B_r(0,PC(H_α)).Fori≥1 andt∈(t_i, t_i+1],we get kΥu(t)k_α ≤ kh_i(t, u(t))−h_i(t,0)k_α+kh_i(t,0)k_α

+ Z ∞

0

ξ_β(θ)kS((t−s_i)^βθ)k kh_i(s_i, u(s_i))−h_i(s_i,0)k_αds (3.8) +

Z ∞ 0

ξβ(θ)kS((t−si)^βθ)kkh_i(si,0)k_αds

+ β

Z t si

Z ∞ 0

θξ_β(θ)(t−s)^β−1kA^αS((t−s)^βθ)k kf(s, u(s), u(g(s)))kdθds

≤ L_hiku(t)k+kh_i(t,0)k_α+M L_hiku(t)k+Mkh_i(t,0)k_α + Cαβ

Z t si

(t−s)^{β(1−α)−1} sup

s∈[si,ti+1],v∈Br(0,PC(Hα))

kf(s, v(s), v(g(s)))kds

×Z ∞ 0

θ^1−αξβ(θ)dθ

≤ L_hi(1 +M)kuk_PC+ (1 +M)kh_i(t,0)k_α + Cα

Γ(2−α) Γ(1 +β(1−α))

T₀^β(1−α)

1−α sup

s∈[s_i,ti+1],v∈Br(0,PC(Hα))

kf(s, v(s), v(g(s)))k

which implies that kΥuk_α ≤r for all i≥1.Similarly, for each t∈[0, t₁],we find that kΥu(t)k_α ≤

Z ∞ 0

ξβ(θ)kS(t^βθ)k kA^αu0kdθ

+ β

Z t 0

Z ∞ 0

θξ_β(θ)(t−s)^β−1kA^αS((t−s)^βθ)kkf(s, u(s), u(g(s)))kdθds

≤ Mku₀k_α+ CαΓ(2−α) Γ(1 +β(1−α))

T₀^β(1−α)

1−α sup

s∈[0,t1],v∈Br(0,PC(Hα))

kf(s, v(s), v(g(s)))k from which we get kΥuk_α ≤r and Υ has values inBr(0,PC(H_α)).

Step 2. The map Υ₁ =PN

i=0Υ¹_i is a contraction on B_r(0,PC(H_α)).

Let t∈(ti, ti+1] and u, v∈Br(0,PC(Hα)), i= 1,· · · , N, we have

kΥ¹_iu(t)−Υ¹_iv(t)k_PC(Hα) ≤(1 +M)L_hiku−vk_{C((ti,ti+1],Hα)}, which implies that kPN

i=0Υ¹_iu−PN

i=0Υ¹_ivk_PC≤Lku−vk_PC, i.e., Υ1 is a contraction on Br(0,PC(H_α)).

Let Υ²_iB_r(0,PC(H_α))(t) ={Υ²_iu(t) :u∈B_r(0,PC(H_α))}.

Step 3. Next, we prove that the set S

Υ²_iBr(0,PC(Hα))(t) is relatively compact inHα.

Let t∈(si, ti+1], for i= 0,1,· · · , N,then for each ∈(si, s) and for each δ >0, we define an operator (Υ²_i)_,δ on B_r(0,PC(H_α)) by

((Υ²_i),δu)(t) = S(^βδ) Z t−

si

(t−s)^β−1 n

β Z ∞

δ

θξβ(θ)S((t−s)^βθ−^βδ)dθ o

× f(s, u(s), u(g(s)))ds, where u∈Br(0,PC(H_α)).The set

(B_r)_,δ(0,PC(H_α))(t) ={((Υ²_i)_,δu)(t) :u∈B_r(0,PC(H_α))}

is relatively compact inHα. Since the operatorS(^βδ), is compact.

Also, for each u∈B_r(0,PC(H_α)), we have k(Υ²_iu)(t)−((Υ²_i)_,δu)(t)k_α

≤ βk Z t

si

Z δ 0

θξ_β(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθds +

Z t si

Z ∞ δ

θξβ(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθds

−

Z t−

si

Z ∞ δ

θξ_β(θ)(t−s)^β−1S((t−s)^βθ)f(s, u(s), u(g(s)))dθdsk_α

≤ β

Z t si

Z δ 0

θξβ(θ)(t−s)^β−1kA^αS((t−s)^βθ)kkf(s, u(s), u(g(s)))kdθds

+ β

Z t t−

Z ∞ δ

θξ_β(θ)(t−s)^β−1kA^αS((t−s)^βθ)kkf(s, u(s), u(g(s)))kdθds

≤ C_αβZ t si

(t−s)^bds1−β1

km_fk

L

1 β1[si,t]

Z δ 0

θ^1−αξ_β(θ)dθ +Cαβ

Z t t−

(t−s)^bds 1−β1

km_fk

L

1 β1[t−,t]

Z ∞ 0

θ^1−αξβ(θ)dθ

≤ βM₁C_αT₀^{(1+b)(1−β1)} (1 +b)^(1−β1)

Z δ 0

θξ_β(θ)dθ+ βM₁C_αΓ(2−α)

Γ(1 +β(1−α))(1 +b)^{(1−β1)(1+b)(1−β1)}. Therefore, there exists relatively compact sets arbitrarily close to the set

SΥ²_iB_r(0,PC(H_α))(t).Thus, the set S

Υ²_iB_r(0,PC(H_α))(t) is relatively compact inH_α. Step 4. In this step, our aim is to prove that the set of functions

[Υ²_iBr(0,^PC(H_α))]i, i= 0,1,· · ·, N is an equicontinuous subset ofC([ti, ti+1], Hα).

Let t₁, t₂∈[s_i, t_i+1], t₁ < t₂ andu∈B_r(0,PC(H_α)), we get k(Υ²_iu)(t₂)−(Υ²_iu)(t₁)k_α

≤ β

Z _t2

t1

Z ∞ 0

θξ_β(θ)(t₂−s)^β−1kA^αS((t₂−s)^βθ)kkf(s, u(s), u(g(s)))kdθds

+ β

Z t1

si

Z ∞ 0

θξ_β(θ)[(t₂−s)^β−1−(t₁−s)^β−1]kA^αS((t₂−s)^βθ)k

×kf(s, u(s), u(g(s)))kdθds

+ β

Z t1

si

Z ∞ 0

θξ_β(θ)(t₁−s)^β−1kA^α[S((t₂−s)^βθ)−S((t₁−s)^βθ)]k

× kf(s, u(s), u(g(s)))kdθds. (3.9)

For the first term on the right hand side of (3.9), we have

β Z t2

t1

Z ∞ 0

θξ_β(θ)(t₂−s)^β−1kA^αS((t₂−s)^βθ)kkf(s, u(s), u(g(s)))kdθds

≤ βC_αM₁Γ(2−α) Γ(1 +β(1−α))

Z t2

t1

(t2−s)^bds 1−β1

≤ βC_αM₁Γ(2−α)

(1 +b)^(1−β1)Γ(1 +β(1−α))(t₂−t₁)^{(1+b)(1−β1)} (3.10) For the second term on the right hand side of (3.9), we have

β Z t1

si

Z ∞ 0

θξβ(θ)[(t2−s)^β−1−(t1−s)^β−1]kA^αS((t2−s)^βθ)k

×kf(s, u(s), u(g(s)))kdθds

≤ βN M1

Γ(1 +β) Z t1

si

[(t2−s)^β−1−(t1−s)^β−1]

1 1−β1ds

1−β1

≤ βN M₁ Γ(1 +β)

Z t1

si

[(t₁−s)^λ−(t₂−s)^λ]ds1−β₁

≤ βN M₁ Γ(1 +β)(1 +λ)^1−β1

(t₂−t₁)^1+λ−((t₂−s_i)^1+λ−(t₁−s_i)^1+λ)1−β1

≤ βN M1

Γ(1 +β)(1 +λ)^1−β1(t2−t1)^{(1+λ)(1−β1)} (3.11)

For t₁ =s_i, it is easy to see that the third term on the right hand side of (3.9) will be zero. Fort₁ > s_i and >0 be sufficiently small, we have

β Z t1−

si

Z ∞ 0

θξ_β(θ)(t₁−s)^β−1kA^α[S((t₂−s)^βθ)−S((t₁−s)^βθ)]k

×kf(s, u(s), u(g(s)))kdθds +β

Z t1

t1−

Z ∞ 0

θξ_β(θ)(t₁−s)^β−1kA^α[S((t₂−s)^βθ)−S((t₁−s)^βθ)]k

×kf(s, u(s), u(g(s)))kdθds

≤ M₁(t^1+λ₁ −^1+λ)^(1−β1) Γ(1 +β)(1 +λ)^1−β1 sup

s∈[si,t1−]

kA^α[S((t₂−s)^βθ)−S((t₁−s)^βθ)]k

+ 2βN M₁

Γ(1 +β)(1 +λ)^{1−β1(1+λ)(1−β1)} (3.12)

Thus, from (3.10)-(3.12) we see that kΥ²_iu(t₂) −Υ²_iu(t₁)k_α tends to zero as t₂ → t₁ for any u ∈ Br(0,PC(H_α)), which means that [Υ²_iBr(0,^PC(H_α))]i is equicontinuous.

Lemma (2.2) and the above steps shows that Υ₁ is a contraction, Υ₂ is completely continuous and Υ = Υ1+ Υ2 is a condensing map onBr(0,PC(Hα)). Then Krasnoselskii’s fixed point theorem ensures that Υ has a fixed point, which gives rise to a mild solution.

4. Application

Consider the following impulsive system of fractional partial differential equations

CD^β_tu(t, x) = ∂²u

∂x² +F(t, x, u(t, x), u(g(t), x)), (t, x)∈

N

[

i=1

[s_i, t_i+1]×[0, π], u(t,0) = u(t, π) = 0, t∈[0, T0]

u(0, x) = u₀(x), x∈[0, π],

u(t, x) = Hi(t, u(t, x)), x∈[0, π], t∈(ti, si]















(4.1)

where 0 = t₀ = s₀ < t₁ ≤ s₁ < · · · < t_N ≤ s_N < t_N+1 = T₀ are fixed real numbers, u₀ ∈ H, F ∈ ([0, T0]×R×R,R) and Hi ∈C((ti, si]×R,R) for all i= 1,· · ·, N.

(A1). Let H=L²([0, π]) andAu=−_∂x^∂22u with D(A) ={u∈H : ∂u

∂x,∂²u

∂x² ∈H, u(0) =u(π) = 0}=H²(0, π)∩H₀¹(0, π),

clearly, the operatorAis the infinitesimal generator of a compact analytic semigroupS(t). Takingα= 1/2, we have D(A^1/2) is Banach space endowed with norm

kuk_1/2 =kA^1/2uk, u∈D(A^1/2).

We can formulate the impulsive system (4.1) in the abstract form (1.4)-(1.6), where u(t) = u(t, .), i.e., u(t)(x) =u(t, x) and the functionsf : [0, T0]×H_1/2×H_1/2→H andhi: (ti, si]×H_1/2→H are given by

f(t, u(t), u(g(t)))(x) = F(t, x, u(t, x), u(g(t), x)), hi(t, u(t))(x) = Hi(t, u(t, x)).

Case 1. Define

(A2). f(t, u(t), u(g(t)))(x) = e^−t{|u(t, x)|+|u(g(t), x)|]}

(γ+e^t){1 +|u(t, x)|+|u(g(t), x)|]}, γ >−1, t∈[0, T₀], u∈H_1/2, x∈(0, π).

Clearly,f : [0, T₀]×H_1/2×H_1/2 →H is continuous function, such that

kf(t, u₁, v₁)−f(t, u₂, v₂)k ≤L_f{ku₁−u₂k_1/2+kv₁−v₂k_1/2}, withL_f = _γ+1¹ .

(A3). hi(t, u(t))(x) = cost|u(t, x)|

2(1 +|u(t, x)|), t∈(ti, si], i= 1,2,· · · , N,

u∈H_1/2, x∈(0, π). (4.2)

Clearly,h_i : (t_i, s_i]×H_1/2 →H are continuous functions, such that kh_i(t, u₁)−f(t, u₂)k ≤L_hiku₁−u₂k_1/2, withL_hi = ¹₂.

(A4). We can chooseg as follows

(i). g(t) =kt for t∈[0, T0], k∈[0,1], (ii). g(t) =ksint for t∈[0, π/2], k∈[0,1].

Hence, (A1)+(A2)+(A3)+(A4) implies that the assumptions in Theorem (3.2) are satisfied. For more details, we refer to [18].

Case 2. Define

(A5). f(t, u(t), u(g(t)))(x) = e^−t(sin(u(t, x)) +cos(u(g(t), x))) (1 +t)(e^t+e^−t) +e^−t, t∈[0, t₁]∪(s₁, t₂]· · · ∪(s_N, T₀], u∈H, x∈(0, π).

Satisfies,

kf(t, u)k ≤ 2e^−t

e^t+e^−t +e^−t=m_f(t), withm_f(t)∈L^∞([0, T0],R+).

Hence, (A1)+(A3)+(A4)+(A5) implies that the assumptions in Theorem (3.3) are also satisfied.

System (4.1) has a mild solution u ∈ PC(H) if u(.) is a mild solution of the associated abstract form (1.4)-(1.6).

The following theorem follows immediately from Theorem 3.2 and Theorem 3.3.

Theorem 4.1. If any of the following assumption is hold. Then there exists a mild solution u∈ PC(H) of (4.1)

(i). The functionsF andH_iare Lipschitz with Lipschitz constantL_F andL_Hi respectively andmax{M L_Hi+

2CαL_FΓ(2−α)

(1−α)Γ(1+β(1−α))T₀^β(1−α): i= 1,· · ·, N}<1.

(ii). The functionsH_i(.)are Lipschitz with Lipschitz constantL_Hi, the functionF(.)is bounded withL_Hi <

1 for alli= 1,· · · , N.

Acknowledgement

We highly appreciate the valuable comments and suggestions of the referees’ on our manuscript which helped to considerably improve the quality of the manuscript. The third author would like to acknowledge the financial aid from the Department of Science and Technology, New Delhi, under its research project SR/S4/MS:796/12.

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